By Lemma 2 and 19 in [1]}, the risk \(R(\widehat{\theta })\)  is evaluated as
\(R(\widehat{\theta }) &\le 2 (\theta ^*)^\top (I - \Pi _\mathsf {Y}) \Sigma (I - \Pi _\mathsf {Y})\theta ^* + \sigma ^2 \mathrm {tr}((\mathsf {Y} \mathsf {Y}^\top )^{-1} \mathsf {Y} \Sigma \mathsf {Y}^\top (\mathsf {Y} \mathsf {Y}^\top )^{-1}) \\& = 2 (\theta ^*)^\top B \theta ^* + c t \sigma ^2 \mathrm {tr}(C),\) 

where \(B= (I - \Pi _\mathsf {Y}) \Sigma (I - \Pi _\mathsf {Y})\) .
We bound the first term as
\((\theta ^*)^\top B \theta ^* &= (\theta ^*)^\top (I - \Pi _\mathsf {Y}) \Sigma (I - \Pi _\mathsf {Y}) \theta ^* \\&=(\theta ^*)^\top (I - \Pi _\mathsf {Y}) (\Sigma - n^{-1} \mathsf {Y}^\top \mathsf {Y}) (I - \Pi _\mathsf {Y}) \theta ^*\\& \le \Vert \theta ^*\Vert ^2 \Vert  I - \Pi _\mathsf {Y}\Vert  \Vert \Sigma - n^{-1} \mathsf {Y} ^\top \mathsf {Y}\Vert  \\& \le \Vert \theta ^*\Vert ^2\Vert \Sigma - n^{-1} \mathsf {Y} ^\top \mathsf {Y}\Vert ,\) 

where the second equality follows \((I - \Pi _\mathsf {Y})\mathsf {Y}^\top = (I - \mathsf {Y}^\top (\mathsf {Y} \mathsf {Y}^\top )^{-1} \mathsf {Y}) \mathsf {Y}^\top = \mathsf {Y}^\top - \mathsf {Y}^\top (\mathsf {Y} \mathsf {Y}^\top )^{-1} (\mathsf {Y} \mathsf {Y}^\top ) = 0\) , and the second inequality follows \(\Vert  I - \Pi _\mathsf {Y}\Vert  \le 1\)  from the non-expansive property of projection operators.
Recalling that \(n^{-1} \mathsf {Y}^\top \mathsf {Y} = \widehat{\Sigma }\)  as in (REF ), Proposition REF  yields the statement.
